RCRC and and L/RL/R Time Constants Time Constants

Topics Covered in Chapter 22 22-1: Response of Resistance Alone

22-2: L/R Time Constant

22-3: High Voltage Produced by Opening an RL Circuit

22-4: RC Time Constant

22-5: RC Charge and Discharge Curves

22-6: High Current Produced by Short-Circuiting RC Circuit

ChapterChapter2222

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Topics Covered in Chapter 22Topics Covered in Chapter 22

22-7: RC Waveshapes 22-8: Long and Short Time Constants 22-9: Charge and Discharge with Short RC Time

Constant 22-10: Long Time Constant for RC Coupling Circuit 22-11: Advanced Time Constant Analysis 22-12: Comparison of Reactance and Time Constant

McGraw-Hill © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

22-1: Response of Resistance 22-1: Response of Resistance AloneAlone

Resistance has only opposition to current. There is no reaction to a change. R has no concentrated magnetic field to oppose a

change in I, like inductance, and no electric field to store charge that opposes a change in V, like capacitance.

22-1: Response of Resistance 22-1: Response of Resistance AloneAlone

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Fig. 22-1:

When the switch S is closed in Fig. 22-1 (a), the battery supplies 10 V across the 10-Ω R and the resultant I is 1 A. The graph in Fig. 22-1 (b) shows that I changes from 0 to 1 A instantly when the switch is closed.

22-2: 22-2: L/RL/R Time Constant Time Constant

The action of an RL circuit during the time current builds up to a specific value is its transient response.

Transient response is a temporary condition that exists only until the steady-state current is reached.

The transient response is measured in terms of the ratio L/R, which is the time constant T of an inductive circuit.

T = L/R The time constant is a measure of how long it takes

the current to change by 63.2%.

22-2: 22-2: L/RL/R Time Constant Time Constant

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Fig. 22-2:

When S is closed, the current changes as I increases from zero. Eventually, I will reach the steady value of 1 A, equal to the battery voltage of 10 V divided by the circuit resistance of 10 Ω.

22-3: High Voltage Produced by 22-3: High Voltage Produced by Opening an Opening an RLRL Circuit Circuit

When an inductive circuit is opened, the time constant for current decay becomes very short because L/R becomes smaller with the high resistance of the open.

Then the current drops toward zero much faster than the rise of current when the switch is closed.

The result is a high value of self-induced voltage across a coil whenever an RL circuit is opened.

This high voltage can be much greater than the applied voltage.

22-3: High Voltage Produced by 22-3: High Voltage Produced by Opening an Opening an RLRL Circuit Circuit

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Fig. 22-3:

In Fig. 22-3, the neon bulb requires 90 V for ionization, at which time it glows. The source is only 8 V, but when the switch is opened, the self-induced voltage is high enough to light the bulb for an instant.The sharp voltage pulse or spike is more than 90 V just after the switch is opened.

22-4:22-4: RC RC Time Constant Time Constant

The transient response of capacitive circuits is measured in terms of the product R x C.

To calculate the time constant,

T = R x C

where R is in ohms, C is in farads, and T is in seconds.

22-4:22-4: RC RC Time Constant Time Constant

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Fig. 22-4:

T = 3 x 106 x 1 x 10−6

= 3 s

22-5:22-5: RC RC Charge and Charge and Discharge CurvesDischarge Curves

Fig. 22-4

In Fig. 22-4, the rise is shown in the RC charge curve because the charging is fastest at the start and then tapers off as C takes on additional charge at a slower rate. As C charges, its potential difference increases. Then the difference in voltage between VT and v C is reduced. Less potential difference reduces the current that puts the charge in C. The more C charges, the more slowly it takes on additional charge.

22-5:22-5: RC RC Charge and Charge and Discharge CurvesDischarge Curves

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T in ms0 1 2 3 4 5

2

4

6

8

0v C

in V

olts

On discharge, C loses its charge at a declining rate. At the start of discharge, vC has its highest value and can produce maximum discharge current. As the discharge continues, vC goes down and there is less discharge current. The more C discharges, the more slowly it loses the remainder of its charge.

22-6: High Current Produced by 22-6: High Current Produced by Short-Circuiting Short-Circuiting RCRC Circuit Circuit

A capacitor can be charged slowly by a small charging current through a high resistance and then be discharged quickly through a low resistance to obtain a momentary surge, or pulse of discharge current.

This idea corresponds to the pulse of high voltage obtained by opening an inductive circuit.

22-6: High Current Produced by 22-6: High Current Produced by Short-Circuiting Short-Circuiting RCRC Circuit Circuit

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Fig. 22-5:

With large capacitors, this can be dangerous!

The circuit of Fig. 22-5 illustrates the application of a battery-capacitor (BC) unit to fire a flashbulb for a camera. The flashbulb needs 5 A to ignite, but this is too much load current for the small 15-V battery. Instead of using the bulb as a load for the battery, the 100-μF capacitor is charged. The capacitor is then discharged through the bulb in Fig. 22-5 (b).

22-7:22-7: RC RC Waveshapes Waveshapes

Voltage and current waveshapes in RC circuits can show when a capacitor is allowed to charge through a resistance for RC time and then discharge through the same resistance for the same amount of time.

Waveshapes show some useful details about the voltage and current for charging and discharging.

22-7:22-7: RC RC Waveshapes Waveshapes

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Fig. 22-6:

22-8: Long and Short Time 22-8: Long and Short Time ConstantsConstants

Useful waveshapes can be obtained by using RC circuits with the required time constant.

In practical applications, RC circuits are used more than RL circuits because almost any value of an RC constant can be obtained easily.

Whether an RC time constant is long or short depends on the pulse width of the applied voltage.

A long time constant can be arbitrarily defined as at least five times longer than the pulse width, in time.

A short time constant is defined as no more than one-fifth the pulse width, in time.

22-8: Long and Short Time 22-8: Long and Short Time ConstantsConstants

VA

R

CvOUT

v OU

T

Integrators use a relatively long time constant.

Differentiators use a relatively short time constant.

VA R

CvOUT

v OU

T

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Integrators and Differentiators

22-9: Charge and Discharge with 22-9: Charge and Discharge with ShortShort RC RC Time ConstantTime Constant

Fig. 22-7

Fig. 22-7 illustrates the charge and discharge of an RC circuit with a short time constant. Note that the waveshape of VR in (d) has sharp voltage peaks for the leading and trailing edges of the square-wave applied voltage.

22-10: Long Time Constant for 22-10: Long Time Constant for RCRC Coupling CircuitCoupling Circuit

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Fig. 22-8:

Fig. 22-8 illustrates the charge and discharge of an RC circuit with a long time constant. Note that the waveshape of VR in (d) has the same waveform as the applied voltage.

22-11: Advanced Time 22-11: Advanced Time Constant AnalysisConstant Analysis

Transient voltage and current values can be determined for any amount of time with a universal time-constant chart.

A universal time-constant chart is a graph of curves obtained by plotting time in RC or L/R time constants versus percent of full voltage or current.

An example of a universal time-constant chart for RC and RL circuits is shown in Fig. 22-9 (next slide).

22-11: Advanced Time 22-11: Advanced Time Constant AnalysisConstant Analysis

.

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Fig. 22-9:

22-12: Comparison of Reactance 22-12: Comparison of Reactance and Time Constantand Time Constant

Table 22-2 Comparison of Reactance XC and RC Time Constant

Sine-Wave Voltage Nonsinusoidal Voltage

Examples are 60-Hz power line, af signal voltage, rf signal voltage

Examples are dc circuit turned on and off, square waves, rectangular pulses

Reactance XC = 1/(2πfC) Time constant T = RC

Larger C results in smaller reactance XC Larger C results in longer time constant

Higher frequency results in smaller XC Shorter pulse width corresponds to longer time constant

IC = VC/XC iC = C(dv/dt)

XC makes IC and VC 90° out of phase Waveshape changes between iC and vC